Short-time-delay limit of the self-coupled FitzHugh-Nagumo system

We analyze the FitzHugh-Nagumo equations subject to time-delayed self-feedback in the activator variable. Parameters are chosen such that the steady state is stable independent of the feedback gain and delay τ. We demonstrate that stable large-amplitude τ-periodic oscillations can, however, coexist with a stable steady state even for small delays, which is mathematically counterintuitive. In order to explore how these solutions appear in the bifurcation diagram, we propose three different strategies. We first analyze the emergence of periodic solutions from Hopf bifurcation points for τ small and show that a subcritical Hopf bifurcation allows the coexistence of stable τ-periodic and stable steady-state solutions. Second, we construct a τ-periodic solution by using singular perturbation techniques appropriate for slow-fast systems. The theory assumes τ=O(1) and its validity as τ→0 is investigated numerically by integrating the original equations. Third, we develop an asymptotic theory where the delay is scaled with respect to the fast timescale of the activator variable. The theory is applied to the FitzHugh-Nagumo equations with threshold nonlinearity, and we show that the branch of τ-periodic solutions emerges from a limit point of limit cycles.